Chapter 2
Double and triple integration
(Stewart (Ed. 7): Chapter 15, p998.)
Chapter Summary
Objective
Double integration over regular domains
Tools
Sketch the domain of integration, decide if the dom
Chapter 1
Partial dierentiation
(Stewart (Ed. 7): Chapter 14, p901.)
Chapter Summary
Objective
Sketch and identify surfaces in 2-D and 3-D.
Find partial derivatives and use these to show simple
result
Thursday, 25th October, 2012
1. Let f (x, y) =
Solutions to 2A (1 and 3) class test
exp(y)
f
f
. Find
and
.
4
2x + y
x
y
5
Solution:
f
0 2 exp(y)
2 exp(y)
=
=
4 )2
x
(2x + y
(2x + y 4 )2
f
(2x + y 4 )
Tuesday, 25th October, 2011
10.10 a.m. to 10.55 a.m.
University of Glasgow
Department of Mathematics
Mathematics 2A - Multivariable Calculus
Class Test
An electronic calculator may be used provided th
Chapter 2
Double and triple integration
Example 2.1 Evaluate
x2 + y 2 dxdy
R
where R is [1, 3] [2, 4].
Solution
The integral may either be evaluated as
3
x2 + y 2 dxdy =
R
4
x2 + y 2 dy
dx
1
2
3
1
x2
Chapter 4
Line and surface integrals: Solutions
Example 4.1 Find the work done by the force F(x, y) = x2 i xyj in moving a particle along the curve
which runs from (1, 0) to (0, 1) along the unit circ
Differentiation
Q1
Pre-course Revision
Find the derivatives of the following functions
a) y = (2x + 3)6 (Ans: 12(2x + 3)5 )
b) y =
c) y =
3
sin(2x )
(Ans: 6cosec(2x )cot(2x ).
1 )
2 x +7
x + 7 (Ans:
x
Chapter 4
Line and surface integrals
Chapter Summary
Objective
Parametric equations of curves
Tools
The position vector of a point on the curve is
(x, y, z) = f (t), t R. This is called a parametric d
Chapter 3
Dierentiation of vectors
Chapter Summary
Objective
Tools
Know and use the denition of grad, div and curl and
understand the meaning of vector and scalar elds
Nabla is the dierential operator
Ex Sheet
5
2A Multivariable Calculus 2014
Tutorial Exercises
.
Lecture 9
Key Points:
Given the change of variables
T1
u=
1
( x + y)
3
denition of a Jacobian
v=
1
( x 2y)
3
use change of variables t
Thursday, 6th December, 2012
4.30 p.m. to 6 p.m.
EXAMINATION FOR THE DEGREES OF
M.A. AND B.Sc.
Mathematics 2A - Multivariable Calculus
An electronic calculator may be used provided that it does not ha
Monday, 12th December, 2011
9.30 a.m. to 11.00 a.m.
2A degree exam - Solutions
1. (i) Solution:
z
z u z v
z 2
z
=
+
=
y exp(xy 2 ) +
.
x
u x v x
u
v
z
z u z v
z
z
z
=
+
=
2yx exp(xy 2 ) +
0=
2yx exp(x
Thursday, 12th December, 2013
1.00 p.m. to 2.30 p.m.
EXAMINATION FOR THE DEGREES OF
M.A. AND B.Sc.
Mathematics 2A - Multivariable Calculus
An electronic calculator may be used provided that it does no
?day, ?th December, 2013
? p.m. to ? p.m.
EXAMINATION FOR THE DEGREES OF
M.A. AND B.Sc.
Mathematics 2A - Multivariable Calculus
An electronic calculator may be used provided that it does not have
a fa
Wednesday, 10th December, 2014
9.30 a.m. to 11 a.m.
EXAMINATION FOR THE DEGREES OF
M.A. AND B.Sc.
Mathematics 2A - Multivariable Calculus
An electronic calculator may be used provided that it does not
Tuesday, 9th November, 2010
Solutions to 2A class test
f
f
and
.
x
y
1. Let f (x, y) = x tan(xy + 2x). Find
5
Solution:
f
xy + 2x
= tan(xy + 2x) +
x
cos2 (xy + 2x)
x2
f
=
y
cos2 (xy + 2x)
2. By making
Chapter 1
Partial dierentiation
Example 1.1 Sketch the graph of f (x, y) = 1 2x x2 y 2 .
Solution
: Let z = f (x, y). Completing the square, we have
z 2 = 1 2x x2 y 2 = 2 (x + 1)2 y 2 ,
i.e. (x + 1)2
5
Ex Sheet
2A Multivariable Calculus 2014
.
Tutorial Exercises
Given the change of variables
T1
u=
1
( x + y)
3
v=
1
( x 2y)
3
express x and y in terms of u and v.
Solution
x = 2u + v, y = u v.
Evalua